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12) Magic squares and cubes: Investigate magic tricks that use mathematics. 13) Loci and complex numbers 14) Egyptian fractions: Egyptian fractions can only have a numerator of 1 – which leads to some interesting patterns. Can all fractions with a numerator of 2 be written as 2 Egyptian fractions?15) Complex numbers and transformations 16) Euler’s identity: An equation that has been voted the most beautiful equation of all time, Euler’s identity links together 5 of the most important numbers in mathematics. This is a puzzle that was posed over 1500 years ago by a Chinese mathematician. 18) Fermat’s last theorem: A problem that puzzled mathematicians for centuries – and one that has only recently been solved.21) Hypercomplex numbers 22) Diophantine application: Cole numbers 23) Perfect Numbers: Perfect numbers are the sum of their factors (apart from the last factor). 24) Euclidean algorithm for GCF 25) Palindrome numbers: Palindrome numbers are the same backwards as forwards.26) Fermat’s little theorem: If p is a prime number then a^p – a is a multiple of p.2) Goldbach’s conjecture: “Every even number greater than 2 can be expressed as the sum of two primes.” One of the great unsolved problems in mathematics.

32) RSA code – the most important code in the world?7) Patterns in Pascal’s triangle: There are a large number of patterns to discover – including the Fibonacci sequence.8) Finding prime numbers: The search for prime numbers and the twin prime conjecture are some of the most important problems in mathematics.The 2 hardest levels – Level 6 and Level 7 are particularly tough – are you good enough to crack them?Ideas for investigation: The authors of the latest Pearson Mathematics SL and HL books have come up with 200 ideas for students doing their maths explorations.